What exactly was Clinton suggesting? That under her tenure the US-UK alliance would be prioritised as it had been in the past? Or was she hedging her bets, affirming the specialness of the relationship in her mind, but implying that this view was not shared by the president or by other members of the US foreign policy establishment?

Statements like Clinton's have set policy-watchers in England on edge. While the US and Britain have long enjoyed a preternaturally intimate bond, some in the UK worry this could change under Obama. His snubbing of the British last summer by choosing Berlin – not London – as his platform to address Europe and the world, and his recent decision to send back a bust of Churchill loaned to the White House after 9/11, have been taken as signs that Obama is not as positively predisposed to the United Kingdom as his predecessors.

But these worries are largely misplaced.

The fact that Gordon Brown is the first EU leader to meet with the president, having narrowly beaten out Nicolas Sarkozy and Angela Merkel for this vaunted diplomatic distinction, speaks to the enduring importance America ascribes to its relationship with Britain.

First and foremost on Brown's and Obama's agenda this week is the resuscitation of the global economy, and on this issue they largely see eye-to-eye. During his visit, Brown is likely to discuss EU efforts to reform international finance – a topic to be treated in detail at next month's G20 summit in London – and his vision for a "global new deal". While the US may prove reluctant to sign on to some of his ambitious global regulatory schemes, Brown should be able to evoke a gesture of support from Obama this week. And on the environment, Obama's recent indication that the US will lead efforts to craft a new global treaty on climate change will surely be greeted warmly by Whitehall.

There is, however, one item on the agenda for Brown's trip that could potentially come in-between the two new friends: Afghanistan.

While the president recently ordered 17,000 more US troops to be sent to combat the resurgent Taliban, and has called upon Nato allies to do similarly, Brown has been reluctant to follow suit. Britain did agree last month to send 300 additional specialist soldiers to the Afghan theatre, but it is unlikely to commit many more. And 10 Downing has made it clear that it will not redeploy its troops currently serving in Basra to Afghanistan following withdrawal from Iraq.

Obama has been right thus far not to condemn Brown's intransigence. The UK troop contingent in Afghanistan, currently at 8,300, is by far the largest European force in the country. And when compared to the total size of the English infantry, only 21,700, this figure is staggering. The UK simply cannot afford to commit any more soldiers without stretching its infantry dangerously thin.

This leaves the US in a bind: Of its Nato allies, England has shouldered the lion's share of responsibility in Afghanistan, but it cannot commit significantly more resources. And of the Nato powers that still can afford to give more, few appear willing – with the notable exception of Italy – to do so. Should Obama push his most committed partner for more, or should he acknowledge that the UK is – in the words of defence secretary John Hutton – already "playing above [its] weight" in Afghanistan, and focus instead on winning over the US's less tractable Nato allies?

By most accounts, Obama is likely to pursue the latter approach. But this does not mean, as some have rashly suggested, that the US can no longer rely on British military support, or that the US-UK security alliance has begun to crack. England will remain America's most serious partner in Afghanistan, and the two countries will continue to share intelligence at a very high level.

Even if, however, the president does call upon the prime minister to commit more troops to Afghanistan, and Brown refuses to march lockstep with Obama as Blair did with Bush, this would not spell the end of their friendship. US-UK relations have always been healthiest when 10 Downing St felt comfortable saying "no" to the White House when pressured to act against its interests. An England secure in its autonomy will remain a steadfast partner of the US simply given the overwhelming convergence of each country's interests and political values. And America, even under the cosmopolitan Obama, will not stray far from its oldest and closest ally.

Doomsayers on each side of the Atlantic should not fret. With the long history of co-operation between the two states, it will take much more than the UK's inability to commit more troops to Central Asia to rent them asunder.